| L(s) = 1 | − 2.64·3-s − 3.47·5-s + 7-s + 4.01·9-s − 3.20·11-s − 1.54·13-s + 9.19·15-s − 17-s + 0.0113·19-s − 2.64·21-s + 6.16·23-s + 7.04·25-s − 2.69·27-s − 5.20·29-s − 2.66·31-s + 8.47·33-s − 3.47·35-s + 3.06·37-s + 4.09·39-s − 2.97·41-s + 7.60·43-s − 13.9·45-s + 4.94·47-s + 49-s + 2.64·51-s − 0.535·53-s + 11.1·55-s + ⋯ |
| L(s) = 1 | − 1.52·3-s − 1.55·5-s + 0.377·7-s + 1.33·9-s − 0.965·11-s − 0.428·13-s + 2.37·15-s − 0.242·17-s + 0.00261·19-s − 0.578·21-s + 1.28·23-s + 1.40·25-s − 0.518·27-s − 0.965·29-s − 0.478·31-s + 1.47·33-s − 0.586·35-s + 0.504·37-s + 0.656·39-s − 0.464·41-s + 1.15·43-s − 2.07·45-s + 0.721·47-s + 0.142·49-s + 0.370·51-s − 0.0735·53-s + 1.49·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 5 | \( 1 + 3.47T + 5T^{2} \) |
| 11 | \( 1 + 3.20T + 11T^{2} \) |
| 13 | \( 1 + 1.54T + 13T^{2} \) |
| 19 | \( 1 - 0.0113T + 19T^{2} \) |
| 23 | \( 1 - 6.16T + 23T^{2} \) |
| 29 | \( 1 + 5.20T + 29T^{2} \) |
| 31 | \( 1 + 2.66T + 31T^{2} \) |
| 37 | \( 1 - 3.06T + 37T^{2} \) |
| 41 | \( 1 + 2.97T + 41T^{2} \) |
| 43 | \( 1 - 7.60T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 + 0.535T + 53T^{2} \) |
| 59 | \( 1 - 8.21T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 - 9.29T + 67T^{2} \) |
| 71 | \( 1 + 1.53T + 71T^{2} \) |
| 73 | \( 1 + 9.37T + 73T^{2} \) |
| 79 | \( 1 - 8.74T + 79T^{2} \) |
| 83 | \( 1 + 1.44T + 83T^{2} \) |
| 89 | \( 1 + 6.23T + 89T^{2} \) |
| 97 | \( 1 + 3.03T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.941271148735698316805286812745, −7.26605820629271744870427113779, −6.84756468064255934341972129224, −5.65205789945871388524645946898, −5.16841840605622511680180735647, −4.46514363856594506460521426989, −3.71213151463447537765040714096, −2.51267750702051329617440175156, −0.905558183652686095937116043584, 0,
0.905558183652686095937116043584, 2.51267750702051329617440175156, 3.71213151463447537765040714096, 4.46514363856594506460521426989, 5.16841840605622511680180735647, 5.65205789945871388524645946898, 6.84756468064255934341972129224, 7.26605820629271744870427113779, 7.941271148735698316805286812745
